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    "# 序列模型\n",
    "`sec_sequence`\n",
    "\n",
    "想象一下有人正在看网飞（Netflix，一个国外的视频网站）上的电影。  \n",
    "一名忠实的用户会对每一部电影都给出评价，毕竟一部好电影需要更多的支持和认可。  \n",
    "然而事实证明，事情并不那么简单，随着时间的推移，人们对电影的看法会发生很大的变化。  \n",
    "事实上，心理学家甚至对这些现象起了名字：  \n",
    "* *锚定*（anchoring）效应：基于其他人的意见做出评价。  \n",
    "  例如，奥斯卡颁奖后，受到关注的电影的评分会上升，尽管它还是原来那部电影。  \n",
    "  这种影响将持续几个月，直到人们忘记了这部电影曾经获得的奖项。  \n",
    "  结果表明（ `Wu.Ahmed.Beutel.ea.2017`），这种效应会使评分提高半个百分点以上。  \n",
    "* *享乐适应*（hedonic adaption）：人们迅速接受并且适应一种更好或者更坏的情况作为新的常态。  \n",
    "  例如，在看了很多好电影之后，人们会强烈期望下部电影会更好。  \n",
    "  因此，在许多精彩的电影被看过之后，即使是一部普通的也可能被认为是糟糕的。  \n",
    "* *季节性*（seasonality）：少有观众喜欢在八月看圣诞老人的电影。  \n",
    "* 有时，电影会由于导演或演员在制作中的不当行为变得不受欢迎。  \n",
    "* 有些电影因为其极度糟糕只能成为小众电影。*Plan9from Outer Space*和*Troll2*就因为这个原因而臭名昭著的。  \n",
    "\n",
    "简而言之，电影评分决不是固定不变的。  \n",
    "因此，使用时间动力学可以得到更准确的电影推荐 `Koren.2009`。  \n",
    "当然，序列数据不仅仅是关于电影评分的。  \n",
    "下面给出了更多的场景。  \n",
    "\n",
    "* 在使用程序时，许多用户都有很强的特定习惯。例如，在学生放学后社交媒体应用更受欢迎。在市场开放时股市交易软件更常用。  \n",
    "* 预测明天的股价要比过去的股价更困难，尽管两者都只是估计一个数字。毕竟，先见之明比事后诸葛亮难得多。 在统计学中，前者（对超出已知观测范围进行预测）称为*外推法*（extrapolation），而后者（在现有观测值之间进行估计）称为*内插法*（interpolation）。  \n",
    "* 在本质上，音乐、语音、文本和视频都是连续的。如果它们的序列被我们重排，那么就会失去原有的意义。比如，一个文本标题“狗咬人”远没有“人咬狗”那么令人惊讶，尽管组成两句话的字完全相同。  \n",
    "* 地震具有很强的相关性，即大地震发生后，很可能会有几次小余震，这些余震的强度比非大地震后的余震要大得多。 事实上，地震是时空相关的，即余震通常发生在很短的时间跨度和很近的距离内。  \n",
    "* 人类之间的互动也是连续的，这可以从微博上的争吵和辩论中看出。  "
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    "## 统计工具\n",
    "\n",
    "处理序列数据需要统计工具和新的深度神经网络架构。    \n",
    "为了简单起见，我们以`fig_ftse100`所示的股票价格（富时100指数）为例。    \n",
    "\n",
    "![近30年的富时100指数](../img/ftse100.png)\n",
    "`fig_ftse100`\n",
    "\n",
    "其中，用$x_t$表示价格，即在*时间步*（time step）   \n",
    "$t \\in \\mathbb{Z}^+$时，观察到的价格$x_t$。  \n",
    "请注意，$t$对于本文中的序列通常是离散的，并在整数或其子集上变化。  \n",
    "假设一个交易员想在$t$日的股市中表现良好，于是通过以下途径预测$x_t$：  \n",
    "\n",
    "$$x_t \\sim P(x_t \\mid x_{t-1}, \\ldots, x_1).$$"
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    "### 自回归模型\n",
    "\n",
    "为了实现这个预测，交易员可以使用回归模型，    \n",
    "例如在`sec_linear_concise`中训练的模型，仅有一个主要问题：输入数据的数量，输入$x_{t-1}, \\ldots, x_1$本身因$t$而异。  \n",
    "也就是说，输入数据的数量这个数字将会随着我们遇到的数据量的增加而增加，因此需要一个近似方法来使这个计算变得容易处理。  \n",
    "本章后面的大部分内容将围绕着如何有效估计$P(x_t \\mid x_{t-1}, \\ldots, x_1)$展开。  \n",
    "简单地说，它归结为以下两种策略。  \n",
    "\n",
    "第一种策略，假设在现实情况下相当长的序列$x_{t-1}, \\ldots, x_1$可能是不必要的，因此我们只需要满足某个长度为$\\tau$的时间跨度，即使用观测序列$x_{t-1}, \\ldots, x_{t-\\tau}$。  \n",
    "当下获得的最直接的好处就是参数的数量总是不变的，至少在$t > \\tau$时如此，这就使我们能够训练一个上面提及的深度网络。这种模型被称为*自回归模型*（autoregressive models），因为它们是对自己执行回归。  \n",
    "\n",
    "第二种策略，如`fig_sequence-model`所示，是保留一些对过去观测的总结$h_t$，\n",
    "并且同时更新预测$\\hat{x}_t$和总结$h_t$。\n",
    "这就产生了基于$\\hat{x}_t = P(x_t \\mid h_{t})$估计$x_t$，以及公式$h_t = g(h_{t-1}, x_{t-1})$更新的模型。\n",
    "由于$h_t$从未被观测到，这类模型也被称为*隐变量自回归模型*（latent autoregressive models）。\n",
    "\n",
    "![隐变量自回归模型](../img/sequence-model.svg)\n",
    "`fig_sequence-model`\n",
    "\n",
    "这两种情况都有一个显而易见的问题：如何生成训练数据？\n",
    "一个经典方法是使用历史观测来预测下一个未来观测。\n",
    "显然，我们并不指望时间会停滞不前。\n",
    "然而，一个常见的假设是虽然特定值$x_t$可能会改变，但是序列本身的动力学不会改变。\n",
    "这样的假设是合理的，因为新的动力学一定受新的数据影响，而我们不可能用目前所掌握的数据来预测新的动力学。\n",
    "统计学家称不变的动力学为*静止的*（stationary）。 \n",
    "因此，整个序列的估计值都将通过以下的方式获得：\n",
    "\n",
    "$$P(x_1, \\ldots, x_T) = \\prod_{t=1}^T P(x_t \\mid x_{t-1}, \\ldots, x_1).$$\n",
    "\n",
    "注意，如果我们处理的是离散的对象（如单词），而不是连续的数字，则上述的考虑仍然有效。\n",
    "唯一的差别是，对于离散的对象，\n",
    "我们需要使用分类器而不是回归模型来估计$P(x_t \\mid  x_{t-1}, \\ldots, x_1)$。"
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    "### 马尔可夫模型\n",
    "\n",
    "回想一下，在自回归模型的近似法中，我们使用$x_{t-1}, \\ldots, x_{t-\\tau}$，而不是$x_{t-1}, \\ldots, x_1$来估计$x_t$。\n",
    "只要这种是近似精确的，我们就说序列满足*马尔可夫条件*（Markov condition）。  \n",
    "特别是，如果$\\tau = 1$，得到一个*一阶马尔可夫模型*（first-order Markov model），  \n",
    "$P(x)$由下式给出：\n",
    "\n",
    "$$P(x_1, \\ldots, x_T) = \\prod_{t=1}^T P(x_t \\mid x_{t-1}) \\text{ 当 } P(x_1 \\mid x_0) = P(x_1).$$\n",
    "\n",
    "当假设$x_t$仅是离散值时，这样的模型特别棒，\n",
    "因为在这种情况下，使用动态规划可以沿着马尔可夫链精确地计算结果。  \n",
    "例如，我们可以高效地计算$P(x_{t+1} \\mid x_{t-1})$：\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "P(x_{t+1} \\mid x_{t-1})\n",
    "&= \\frac{\\sum_{x_t} P(x_{t+1}, x_t, x_{t-1})}{P(x_{t-1})}\\\\\n",
    "&= \\frac{\\sum_{x_t} P(x_{t+1} \\mid x_t, x_{t-1}) P(x_t, x_{t-1})}{P(x_{t-1})}\\\\\n",
    "&= \\sum_{x_t} P(x_{t+1} \\mid x_t) P(x_t \\mid x_{t-1})\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "利用这一事实，我们只需要考虑过去观察中的一个非常短的历史：$P(x_{t+1} \\mid x_t, x_{t-1}) = P(x_{t+1} \\mid x_t)$。  \n",
    "隐马尔可夫模型中的动态规划超出了本节的范围 （我们将在`sec_bi_rnn`再次遇到），而动态规划这些计算工具已经在控制算法和强化学习算法广泛使用。  "
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    "### 因果关系\n",
    "\n",
    "原则上，将$P(x_1, \\ldots, x_T)$倒序展开也没什么问题，毕竟，基于条件概率公式，我们总是可以写出：\n",
    "\n",
    "$$P(x_1, \\ldots, x_T) = \\prod_{t=T}^1 P(x_t \\mid x_{t+1}, \\ldots, x_T).$$\n",
    "\n",
    "事实上，如果基于一个马尔可夫模型，我们还可以得到一个反向的条件概率分布。  \n",
    "然而，在许多情况下，数据存在一个自然的方向，即在时间上是前进的。    \n",
    "很明显，未来的事件不能影响过去。因此，如果我们改变$x_t$，可能会影响未来发生的事情$x_{t+1}$，但不能反过来。也就是说，如果我们改变$x_t$，基于过去事件得到的分布不会改变。  \n",
    "\n",
    "因此，解释$P(x_{t+1} \\mid x_t)$应该比解释$P(x_t \\mid x_{t+1})$更容易。\n",
    "例如，在某些情况下，对于某些可加性噪声$\\epsilon$，显然我们可以找到$x_{t+1} = f(x_t) + \\epsilon$，而反之则不行`Hoyer.Janzing.Mooij.ea.2009`。  \n",
    "而这个向前推进的方向恰好也是我们通常感兴趣的方向。彼得斯等人`Peters.Janzing.Scholkopf.2017`对该主题的更多内容做了详尽的解释，而我们的上述讨论只是其中的冰山一角。"
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    "\n",
    "## 训练\n",
    "\n",
    "在了解了上述统计工具后，让我们在实践中尝试一下！\n",
    "首先，我们生成一些数据：(**使用正弦函数和一些可加性噪声来生成序列数据，时间步为$1, 2, \\ldots, 1000$。**)\n"
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   "cell_type": "code",
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   "source": [
    "import os\n",
    "import sys\n",
    "\n",
    "current_directory = os.getcwd()\n",
    "parent_directory = os.path.dirname(current_directory)\n",
    "if parent_directory not in sys.path:\n",
    "    sys.path.append(parent_directory)"
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  {
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   "execution_count": 2,
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    "origin_pos": 2,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8703181c",
   "metadata": {
    "origin_pos": 5,
    "tab": [
     "pytorch"
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   },
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       "<Figure size 600x300 with 1 Axes>"
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   "source": [
    "T = 1000  # 总共产生1000个点\n",
    "time = torch.arange(1, T + 1, dtype=torch.float32)\n",
    "x = torch.sin(0.01 * time) + torch.normal(0, 0.2, (T,))\n",
    "d2l.plot(time, [x], 'time', 'x', xlim=[1, 1000], figsize=(6, 3))"
   ]
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   "cell_type": "markdown",
   "id": "91a4997c",
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   "source": [
    "接下来，我们将这个序列转换为模型的*特征－标签*（feature-label）对。   \n",
    "基于嵌入维度$\\tau$，我们[**将数据映射为数据对$y_t = x_t$ 和 $\\mathbf{x}_t = [x_{t-\\tau}, \\ldots, x_{t-1}]$。**]\n",
    "这比我们提供的数据样本少了$\\tau$个， \n",
    "因为我们没有足够的历史记录来描述前$\\tau$个数据样本。    \n",
    "一个简单的解决办法是：如果拥有足够长的序列就丢弃这几项；另一个方法是用零填充序列。  \n",
    "在这里，我们仅使用前600个“特征－标签”对进行训练。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "15071cc2",
   "metadata": {
    "origin_pos": 8,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "tau = 4\n",
    "features = torch.zeros((T - tau, tau))\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i: T - tau + i]\n",
    "labels = x[tau:].reshape((-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "de5cf9b0",
   "metadata": {
    "origin_pos": 10,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "batch_size, n_train = 16, 600\n",
    "# 只有前n_train个样本用于训练\n",
    "train_iter = d2l.load_array((features[:n_train], labels[:n_train]),\n",
    "                            batch_size, is_train=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f830b1f",
   "metadata": {
    "origin_pos": 11
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   "source": [
    "在这里，我们[**使用一个相当简单的架构训练模型：\n",
    "一个拥有两个全连接层的多层感知机**]，ReLU激活函数和平方损失。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "bf897572",
   "metadata": {
    "origin_pos": 13,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "# 初始化网络权重的函数\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        nn.init.xavier_uniform_(m.weight)\n",
    "\n",
    "# 一个简单的多层感知机\n",
    "def get_net():\n",
    "    net = nn.Sequential(nn.Linear(4, 10),\n",
    "                        nn.ReLU(),\n",
    "                        nn.Linear(10, 1))\n",
    "    net.apply(init_weights)\n",
    "    return net\n",
    "\n",
    "# 平方损失。注意：MSELoss计算平方误差时不带系数1/2\n",
    "loss = nn.MSELoss(reduction='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62592eee",
   "metadata": {
    "origin_pos": 16
   },
   "source": [
    "现在，准备[**训练模型**]了。实现下面的训练代码的方式与前面几节（如`sec_linear_concise`）中的循环训练基本相同。因此，我们不会深入探讨太多细节。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8842e1a5",
   "metadata": {
    "origin_pos": 18,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss: 0.062940\n",
      "epoch 2, loss: 0.057493\n",
      "epoch 3, loss: 0.053914\n",
      "epoch 4, loss: 0.053884\n",
      "epoch 5, loss: 0.054631\n"
     ]
    }
   ],
   "source": [
    "def train(net, train_iter, loss, epochs, lr):\n",
    "    trainer = torch.optim.Adam(net.parameters(), lr)\n",
    "    for epoch in range(epochs):\n",
    "        for X, y in train_iter:\n",
    "            trainer.zero_grad()\n",
    "            l = loss(net(X), y)\n",
    "            l.sum().backward()\n",
    "            trainer.step()\n",
    "        print(f'epoch {epoch + 1}, '\n",
    "              f'loss: {d2l.evaluate_loss(net, train_iter, loss):f}')\n",
    "\n",
    "net = get_net()\n",
    "train(net, train_iter, loss, 5, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e60966e7",
   "metadata": {
    "origin_pos": 21
   },
   "source": [
    "## 预测\n",
    "\n",
    "由于训练损失很小，因此我们期望模型能有很好的工作效果。  \n",
    "让我们看看这在实践中意味着什么。  \n",
    "首先是检查[**模型预测下一个时间步**]的能力，也就是*单步预测*（one-step-ahead prediction）。  \n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "id": "c8fc264a",
   "metadata": {
    "origin_pos": 22,
    "tab": [
     "pytorch"
    ]
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   "outputs": [
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   "source": [
    "onestep_preds = net(features)\n",
    "d2l.plot([time, time[tau:]],\n",
    "         [x.detach().numpy(), onestep_preds.detach().numpy()], 'time',\n",
    "         'x', legend=['data', '1-step preds'], xlim=[1, 1000],\n",
    "         figsize=(6, 3))"
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   "cell_type": "markdown",
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    "正如我们所料，单步预测效果不错。  \n",
    "即使这些预测的时间步超过了$600+4$（`n_train + tau`），其结果看起来仍然是可信的。  \n",
    "然而有一个小问题：如果数据观察序列的时间步只到$604$，  \n",
    "我们需要一步一步地向前迈进：  \n",
    "$$\n",
    "\\hat{x}_{605} = f(x_{601}, x_{602}, x_{603}, x_{604}), \\\\\n",
    "\\hat{x}_{606} = f(x_{602}, x_{603}, x_{604}, \\hat{x}_{605}), \\\\\n",
    "\\hat{x}_{607} = f(x_{603}, x_{604}, \\hat{x}_{605}, \\hat{x}_{606}),\\\\\n",
    "\\hat{x}_{608} = f(x_{604}, \\hat{x}_{605}, \\hat{x}_{606}, \\hat{x}_{607}),\\\\\n",
    "\\hat{x}_{609} = f(\\hat{x}_{605}, \\hat{x}_{606}, \\hat{x}_{607}, \\hat{x}_{608}),\\\\\n",
    "\\ldots\n",
    "$$\n",
    "\n",
    "通常，对于直到$x_t$的观测序列，其在时间步$t+k$处的预测输出$\\hat{x}_{t+k}$  \n",
    "称为$k$*步预测*（$k$-step-ahead-prediction）。  \n",
    "由于我们的观察已经到了$x_{604}$，它的$k$步预测是$\\hat{x}_{604+k}$。  \n",
    "换句话说，我们必须使用我们自己的预测（而不是原始数据）来[**进行多步预测**]。  \n",
    "让我们看看效果如何。  \n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "89ba8713",
   "metadata": {
    "origin_pos": 24,
    "tab": [
     "pytorch"
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   },
   "outputs": [],
   "source": [
    "multistep_preds = torch.zeros(T)\n",
    "multistep_preds[: n_train + tau] = x[: n_train + tau]\n",
    "for i in range(n_train + tau, T):\n",
    "    multistep_preds[i] = net(\n",
    "        multistep_preds[i - tau:i].reshape((1, -1)))"
   ]
  },
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   "id": "f29da1fa",
   "metadata": {
    "origin_pos": 27,
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    "d2l.plot([time, time[tau:], time[n_train + tau:]],\n",
    "         [x.detach().numpy(), onestep_preds.detach().numpy(),\n",
    "          multistep_preds[n_train + tau:].detach().numpy()], 'time',\n",
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    "如上面的例子所示，绿线的预测显然并不理想。  \n",
    "经过几个预测步骤之后，预测的结果很快就会衰减到一个常数。  \n",
    "为什么这个算法效果这么差呢？事实是由于错误的累积：  \n",
    "假设在步骤$1$之后，我们积累了一些错误$\\epsilon_1 = \\bar\\epsilon$。\n",
    "于是，步骤$2$的输入被扰动了$\\epsilon_1$，  \n",
    "结果积累的误差是依照次序的$\\epsilon_2 = \\bar\\epsilon + c \\epsilon_1$，\n",
    "其中$c$为某个常数，后面的预测误差依此类推。  \n",
    "因此误差可能会相当快地偏离真实的观测结果。  \n",
    "例如，未来$24$小时的天气预报往往相当准确，  \n",
    "但超过这一点，精度就会迅速下降。  \n",
    "我们将在本章及后续章节中讨论如何改进这一点。  \n",
    " \n",
    "基于$k = 1, 4, 16, 64$，通过对整个序列预测的计算，让我们[**更仔细地看一下$k$步预测**]的困难。  \n"
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   "cell_type": "code",
   "execution_count": 10,
   "id": "86fd62af",
   "metadata": {
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     "pytorch"
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   "source": [
    "max_steps = 64"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b5468e42",
   "metadata": {
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   "source": [
    "features = torch.zeros((T - tau - max_steps + 1, tau + max_steps))\n",
    "# 列i（i<tau）是来自x的观测，其时间步从（i）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i: i + T - tau - max_steps + 1]\n",
    "\n",
    "# 列i（i>=tau）是来自（i-tau+1）步的预测，其时间步从（i）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau, tau + max_steps):\n",
    "    features[:, i] = net(features[:, i - tau:i]).reshape(-1)"
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   "id": "e41dbccd",
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      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
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   ],
   "source": [
    "steps = (1, 4, 16, 64)\n",
    "d2l.plot([time[tau + i - 1: T - max_steps + i] for i in steps],\n",
    "         [features[:, (tau + i - 1)].detach().numpy() for i in steps], 'time', 'x',\n",
    "         legend=[f'{i}-step preds' for i in steps], xlim=[5, 1000],\n",
    "         figsize=(6, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13e7b084",
   "metadata": {
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   "source": [
    "以上例子清楚地说明了当我们试图预测更远的未来时，预测的质量是如何变化的。\n",
    "虽然“$4$步预测”看起来仍然不错，但超过这个跨度的任何预测几乎都是无用的。\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e188c3ed-d82c-491e-a264-c6be3eb36f16",
   "metadata": {},
   "source": [
    "## 小结\n",
    "\n",
    "* 内插法（在现有观测值之间进行估计）和外推法（对超出已知观测范围进行预测）在实践的难度上差别很大。因此，对于所拥有的序列数据，在训练时始终要尊重其时间顺序，即最好不要基于未来的数据进行训练。\n",
    "* 序列模型的估计需要专门的统计工具，两种较流行的选择是自回归模型和隐变量自回归模型。\n",
    "* 对于时间是向前推进的因果模型，正向估计通常比反向估计更容易。\n",
    "* 对于直到时间步$t$的观测序列，其在时间步$t+k$的预测输出是“$k$步预测”。随着我们对预测时间$k$值的增加，会造成误差的快速累积和预测质量的极速下降。"
   ]
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   "id": "028b5014-9150-46a5-8a39-a9725399076a",
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   "source": []
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